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arXiv:2402.18531

Phase transitions beyond criticality: extending Ising universal scaling functions to describe entire phases

David J. Hathcock, James P. Sethna

Published Feb 28, 2024• 0 citations
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Abstract

Universal scaling laws only apply asymptotically near critical phase transitions. We propose a general scheme, based on normal form theory of renormalization group flows, for incorporating corrections to scaling that quantitatively describe the entire neighboring phases. Expanding Onsager's exact solution of the 2D Ising model about the critical point, we identify a special coordinate with radius of convergence covering the entire physical temperature range, $0

DOI
10.48550/arxiv.2402.18531

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Universal scaling laws only apply asymptotically near critical phase transitions. We propose a general scheme, based on normal form theory of renormalization group flows, for incorporating corrections to scaling that quantitatively describe the entire neighboring phases. Expanding Onsager's exact solution of the 2D Ising model about the critical point, we identify a special coordinate with radius of convergence covering the entire physical temperature range, $0
Our community has a deep and sophisticated understanding of phase transitions and their universal scaling functions. We outline and advocate an ambitious program to use this understanding as an anchor for describing the surrounding phases. We explain how to use normal form theory to write universal scaling functions in systems where the renormalization-group flows cannot be linearized. We use the 2d Ising model to demonstrate how to calculate high-precision implementations of universal scaling functions, and how to extend them into a complete description of the surrounding phases. We discuss prospects and challenges involved into extending these early successes to the many other systems where the RG has successfully described emergent scale invariance, making them invaluable tools for engineers, biologists, and social scientists studying complex systems.
The results of the renormalization group are commonly advertised as the existence of power law singularities near critical points. Logarithmic and exponential corrections are seen as special cases and dealt with on a case-by-case basis. We propose to systematize the analysis of singularities in the renormalization group using perturbative normal form theory. Classification of all such singularities in this unified framework generates a systematic machinery to perform scaling collapses. We show that this procedure leads to a better handling of the singularity even in the classic case of the 4-d Ising model.
The results of the renormalization group are commonly advertised as the existence of power-law singularities near critical points.The classic predictions are often violated and logarithmic and exponential corrections are treated on a case-by-case basis.We use the mathematics of normal form theory to systematically group these into universality families of seemingly unrelated systems united by common scaling variables.We recover and explain the existing literature and predict the nonlinear generalization for the universal homogeneous scaling functions.We show that this procedure leads to a better handling of the singularity even in classic cases and elaborate our framework using several examples.
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Renormalization group theory is a powerful and intriguing technique with a wide range of applications. One of the main successes of renormalization group theory is the description of continuous phase transitions and the development of scaling theory. Most courses on phase transitions focus on scaling and critical exponents, while less attention is paid to universality, renormalization group flow, and the existence of a unique fixed point, which are the ultimate reasons why scaling theory is so effective in describing continuous phase transitions. We use a combination of Monte Carlo simulations and real space renormalization group theory to determine the renormalization group flow and to show the existence of a universal fixed point in the context of the ferromagnetic Ising model.